Der_CH05 (1) ACtuarial

8/22/2019 Der_CH05 (1) ACtuarial

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5-1

Financial Forwards & Futures


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5-2

Pricing Prepaid Forwards

If we can price theprepaidforward (FP),then we can calculate the forward price(F) for a regular forward contract

F= FV(FP)

For now, assume that there are no

dividends


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5-3

Pricing Prepaid Forwards with arbitrage

Arbitrage: a situation in which one can generate positive cash

flow by simultaneously buying and selling related assets, withno net investment and with no risk

If, at time t=0, the prepaid forward price somehow exceededthe stock price, i.e., , an arbitrageur could do thefollowing:

Since arbitrage profits are traded away quickly, in equilibriumwe expect:

FP

0,T S0

FP

0,T S0


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5-4

Pricing Prepaid Forwards (contd)

What if there are dividends? Isstill valid?

No, because the holder of the forward will not

receive dividends that will be paid to the holder ofthe stockso you will not pay for it in forward mrkt

For discrete dividends Dtiat times ti, i= 1,., n The prepaid forward price: For continuous dividends with an annualized yield d

The prepaid forward price:

FP0,T S0 ni1PV0,ti (Dti )

FP

0,T S0edT

FP

0,T S0 PV (all dividends paid from t=0 to t=T)

FP

0,T S0


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5-5

Example 5.1

XYZ stock costs $100 today and isexpected to pay a quarterly dividend of$1.25. If the risk-free rate is 10%compounded continuously, how muchdoes a 1-year prepaid forward cost?


Fp0,1 $100 4 i1$1.25e

0.025i $95.30


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5-6

Example 5.2

The index is $125 and the dividend yieldis 3% continuously compounded. Howmuch does a 1-year prepaid forward cost?


FP0,1 $125e0.03

$121.31


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5-7

Pricing Forwards on Stock

Forward price is the future value oftheprepaidforward

No dividends:

Discrete dividends:

Continuous dividends: TrT eSF )(0,0 d

F0,T S0erT

n

i1er(Tti )Dti

rT

TeSF

0,0


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5-8

Forward when underlying pays discretedividends (alt question 5.2)

A $50 stock pays $1 dividend every 3months, with the next in exactly 3months. What is the 1 year forwardprice if the continuously compoundedrate is 6%?


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5-9

Forward premium

Premium = the difference betweencurrent forward price and stock price

Can be used to infer the current stock pricefrom forward price

Definition

Forward premium = F0, T / S0 Annualized forward premium =

(1/T) ln (F0, T / S0)


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5-10

Another synthetic forward:

Payoffs:Step Today time=t

Borrow S +S

Buy and Sell Asset -S +S(T)

Repay S*(1+r) -S(1+r)

Net Investment/Payoff 0 S(T)-S(1+r)

This replicates a long forward, right?


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5-11

Creating a SyntheticForward

One can offset the risk of a forward by creating asyntheticforward to offset a position in the actualforward contract

How can one do this? (assume continuous dividends at rate d)

Recall the long forward payoff at expiration: = STF0, T Borrow and purchase shares as follows

Note that the total payoff at expiration is same as

forward payoff

A tailedposition


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5-12

Creating a SyntheticForward (contd)

The idea of creating synthetic forward leads to following

Forward = Stock zero-coupon bond (buy index on borrowed $)

Stock = Forward zero-coupon bond

Zero-coupon bond = Stock forward

Cash-and-carry arbitrage: Buy the index, short the forward

Arbitrage proof of pricing relation


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5-13

Forward-spot no-arbitrage, again

Imagine F = $20, but S=$10 and r=10% effectiveannual yield.

I could borrow $10 and buy the asset, and thensimultaneously enter short forward contract.

At maturity,

I sell at F=$20, and repay $11, earning a risk-freereturn with zero net investment.

To eliminate such opportunities, it must be thatF = the cost of carry

The cost of carry often includes dividends (bonds, currencies,

equities), storage costs and convenience yields (commodities) Above the cost of carry is S(1+r), therefore,

F = S(1+r)


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5-14

Forward-Spot No-Arbitrage, again Imagine F = $20, but S=$10, r=10%, and it will pay a $1 dividend

in one year. I could borrow $10 and buy the asset, and then simultaneously enter

short forward contract.

At maturity,

I sell at F=$20, and repay $11 and keep $1 dividend

The dividend lowers the cost of carry To eliminate such opportunities, it must be that F = the cost of carry

Here the cost of carry = FV(S) - FV(dividend)

F = 10(1.1) -1 = 10

To conduct arbitrage:Step Today 1 Year

Borrow S 10

Buy Asset -10

Receive Dividend 1

Payback Loan -11

Sell Forward 0 10

SUM 0 0


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5-15

Other Issues in Forward Pricing

Does the forward price predict thefuture price?

According the formula the forwardprice conveys no additional information beyond what S0 ,r, and dprovides

The forward price does converge to the spot price asmaturity approaches

Forward pricing formula and cost of carry

Forward price =Spot price + Interest to carry the asset asset lease rate

Cost of carry, (r-d)S

Tr

T eSF)(

0,0

d


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5-16

Futures Contracts Futures are exchange-traded forward contracts

Typical features of futures contracts

Standardized, with specified delivery dates, locations, procedures

A clearinghouse

Matches buy and sell orders

Keeps track of members obligations and payments

After matching the trades, becomes counterparty

Differences from forward contracts

Settled daily through the mark-to-market process low credit risk

Highly liquid easier to offset an existing position

Highly standardized structure harder to customize


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5-17

Example: S&P 500 Futures

Notional value: $250 x Index

Cash-settled contract

Open interest: total number of buy/sell pairs

Margin and mark-to-market

Initial margin (5-10%)

Maintenance margin (70 80% of initial margin)

Daily mark-to-market


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5-18

Example: S&P 500 Futures (contd)

Futures prices versus forward prices

The difference is negligible especially forshort-lived contracts

Small differences due to time-value of margingain/losses

Can be significant for long-lived contractswhen forward contracts contain greaterdefault premia


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5-19

Example: S&P 500 Futures (contd) Mark-to-market proceeds and margin

balance for 8 long futures

10% initial margin, r=6%=.1(250*8*F)

If forward contract, 1011.65-1100=-88.35 or -176700

Implies

loss of

$177,562


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5-20

Uses of Index Futures Why buy an index futures contract instead of synthesizing it using the stocks in the index?

Lower transaction costs Asset allocation/ Hedging: switching investments among asset classes

Example: invested in the S&P 500 index and temporarily wish to invest inbonds instead of index. What to do?

Alternative #1: sell all 500 stocks and invest in bonds

Alternative #2: take a short forward position in S&P 500 index and keepyour stock position

You converted your stock investment into a bond


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5-21

A hedging example

Suppose we invest $Ip

in a portfolio ofSP500 stocks.

Recall:

What is in this case?

Since my asset is SP500 and there are SP500futures, this is a direct hedge: =1.

But dont short one contract, we need to

adjust for portfolio size Let N=notional value of forward contract,

then

*H

N

IH

p*


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5-22

Example continued

Suppose we have $1,200,000 invested in theSP500. The SP500 futures contracts arepriced at 1270e(.04-.02)1=1295.65, and eachcontract has notional value of 250*S0.

Solve for H* Ip = $1,200,000, N= 250*1270=$317,500

H*=-{1200000/(317,500)}(1)

H*=-3.78 which means we short 3.78 contracts

Note: weve invested in 1,200,000/1270=944.88 shares

Directhedge


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5-23

Lets suppose we short 4 contracts

Outcome of hedge? Payoff = ST*944.88 +4*250*(1295.65-ST)

What if ST = 0?

Payoff = 0 +1,295,650 = 1,295,650

What if ST= 2000?

Payoff = 1,889,760+4(250)(-704.35)

Payoff = 1,889,760-704,350 = 1,185,410

Its not equal, what happened? We overhedged (on short side we did better

when prices fell)


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5-24

What if we did short 3.78 contracts?

Outcome of hedge? Payoff = ST*944.88 +3.78*250*(1295.65-ST)

What if ST = 0? Payoff = 0 + 1,224,389 = 1,224,389

What if ST= 2000? Payoff = 1,889,760+3.78(250)(-704.35)

Payoff = 1,889,760- 665,610 = 1,224,150

Rounding errors

Note, I end with more than I started, indeedI earned the risk-free rate

See Table 5.10 (and discussion thereof)


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5-25

Currency Contracts: Pricing

Currency prepaid forward

Suppose you want to purchase 1 one year from todayusing $s

Wherex0 is current ($/ ) exchange rate, and ry is theyen-denominated interest rate

Why? By deferring exchange of the currency one loses interestincome from deposits denominated in that currency

Currency forward

ris the $-denominated domestic interest rate

F0, T>x0 ifr> ry(domestic risk-free rate exceedsforeign risk-free rate), and vice-versa

FP0,T x0eryT

F0,T x0e(rry )t


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5-26

Currency Contracts: Pricing (contd)

Example 5.3

-denominated interest rate is 2% and current ($/ )exchange rate is 0.009 $/ . To have 1 in one year, oneneeds to invest today

0.009$/ x 1 x e-0.02

= $0.008822 Check:

$.008822*(1/.009)*e0.02= 1

Example 5.4

-denominated interest rate is 2% and $-denominatedrate is 6%. The current ($/ ) exchange rate is 0.009.The 1-year forward rate

0.009e0.06-0.02 = 0.009367


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5-27

Table 5.12

Currency Contracts: Pricing (contd)

Synthetic currency forward: borrowing in one currencyand lending in another creates the same cash flow asa forward contract

Covered interest arbitrage: offset the synthetic forwardposition with an actual forward contract

Now,

imagine if

F = .01?

Short actual forward and lock in $0 000633

$(xT)-$0.009367

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Der_CH05 (1) ACtuarial